Let $G$ be a group.
Then, $G$ is free iff there exists $X\subset G$ such that for any group $H$ and a function $f\in H^X$, there exists a unique $\phi\in Hom(G,H)$ such that $\phi\upharpoonright X=f$. It can be shown that if $G$ is a group and $X\subseteq G$, then $G$ is free with base $X$ if and only if every $g\in G$ can be uniquely written in the form $g=x_1 x_2\ldots x_n$ where $0\leq n$, $x_i\in X\cup X^{-1}$ for all $1\leq i\leq n$ and $x_{i+1}\neq x_i$ for all $1\leq i\leq n-1$.
Question. I know that $G$ is finitely generated free group with generating set $S$, $G=\langle S\rangle$. Is $G$, free with base $S$? ($S$ is symetric)